sols2 - 2. SEPTEMBER 1ST: LOGIC 3 2. September 1st: Logic...

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2. SEPTEMBER 1ST: LOGIC 3 2. September 1st: Logic 2.1 . Prove De Morgan’s laws. Answer: Recall that these say ¬ ( p q ) ⇐⇒ ( ¬ p ) ( ¬ q ) ¬ ( p q ) ( ¬ p ) ( ¬ q ) . We can take care of these with two truth tables: p q ¬ p ¬ q p q ¬ ( p q ) ( ¬ p ) ( ¬ q ) T T F F T F F T F F T T F F F T T F T F F F F T T F T T proves the ±rst law, and p q ¬ p ¬ q p q ¬ ( p q ) ( ¬ p ) ( ¬ q ) T T F F T F F T F F T F T T F T T F F T T F F T T F T T proves the second. ± Note that the second law actually follows from the ±rst (or conversely, if you prefer), so one table would su²ce. To see this, assume you have proved the ±rst law; negate both sides to get p q ⇐⇒ ¬ (( ¬ p ) ( ¬ q )) ; then apply this tautology to p = ¬ r and q = ¬ s : ( ¬ r ) ( ¬ s ) ( r s ) . Of course r and s are just names for arbitrary statements, so the last tautology we obtained is just a restatement of the second law. 2.2 . Assume that L ( x ) means x is a lion’; C ( x ) means x is a cow’; R ( x ) means x is red’; G ( x ) means: x
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This note was uploaded on 12/14/2011 for the course MGF 3301 taught by Professor Aluffi during the Fall '11 term at FSU.

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sols2 - 2. SEPTEMBER 1ST: LOGIC 3 2. September 1st: Logic...

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